Propagation of waves

1. Propagation of waves Friday October 18, 2002

2. Propagation of waves in 3D • Imagine a disturbane that results in waves propagating equally in all directions • E.g. sound wave source in air or water, light source in a dielectric medium etc.. • The generalization of the wave equation to 3-dimensions is straight forward if the medium is homogeneous • Let  = amplitude of disturbance (could be amplitude of E-field also)

3. Propagation of waves in 3D  depends on x, y and z such that it satisfies the wave equation or, where in cartesian co-ordinates,

4. 1. Special Case: Plane Waves along x • Suppose (x, y, z, t)=(x, t) (depends only on x) • Then  = f(kx-ωt) + g(kx+ωt) • Then for a given position xo,  has the same value for all y, z at any time to. • i.e. the disturbance has the same value in the y-z plane that intersects the x-axis at xo. • This is a surface of constant phase

5. Plane waves along x Planes perpendicular to the x-axis are wave fronts – by definition

6. z O y x 2. Plane waves along an arbitrary direction (n) of propagation • Now  will be constant in plane perpendicular to n – if wave is plane • For all points P’ in plane P P’ d

7. 2. Plane waves along an arbitrary direction (n) of propagation For all points P’ in plane or, for the disturbance at P

8. z O y x 2. Plane waves along an arbitrary direction (n) of propagation If wave is plane,  must be the same everywhere in plane  to n This plane is defined by P P’ d is equation of a plane  to n, a distance d from the origin

9. 2. Plane waves along an arbitrary direction (n) of propagation is the equation of a plane wave propagating in k-direction

10. 3. Spherical Waves • Assume has spherical symmetry about origin (where source is located) • In spherical polar co-ordinates z θ r y φ x

11. 3. Spherical Waves • Given spherical symmetry,  depends only on r, not φ or θ • Consequently, the wave equation can be written,

12. 3. Spherical Waves Now note that,

13. 3. Spherical Waves But, is just the wave equation, whose solution is, i.e. amplitude decreases as 1/ r !! Wave fronts are spheres

14. 4. Cylindrical Waves (e.g. line source) The corresponding expression is, for a cylindrical wave traveling along positive 

15. Electromagnetic waves • Consider propagation in a homogeneous medium (no absorption) characterized by a dielectric constant o = permittivity of free space

16. Electromagnetic waves Maxwell’s equations are, in a region of no free charges, Gauss’ law – electric field from a charge distribution No magnetic monopoles Electromagnetic induction (time varying magnetic field producing an electric field) Magnetic fields being induced By currents and a time-varying electric fields µo = permeability of free space (medium is diamagnetic)

17. Electromagnetic waves For the electric field E, or, i.e. wave equation with v2 = 1/µo

18. Electromagnetic waves Similarly for the magnetic field i.e. wave equation with v2 = 1/µo In free space,  =  o = o ( = 1) c = 3.0 X 108 m/s

19. Electromagnetic waves In a dielectric medium,  = n2 and =  o = n2 o

20. Electromagnetic waves: Phase relations The solutions to the wave equations, can be plane waves,